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In recent years, tremendous advances in communication technologies coupled with the advent of the Internet of Things (IoT) have led to the emergence of the Big Data phenomenon. Big Data is one of the big IT challenges of the current decade. The amount of data produced is constantly increasing and makes it more and more difficult to process. Managing these masses of data requires the use of new data management systems with efficient access methods. Considered as one of the main sources of Big Data, wireless sensors used in networks offer a credible solution to the problem of Big Data management, especially its collection. Several solutions for Big Data collection based on large-scale wireless sensor networks (LS-WSN) are proposed, taking into account the nature of the applications. The hierarchical architecture is the one used for the deployment of these applications. In such an architecture, relay sensors play an important role in finding the balance of the network and maximizing its lifetime. In most LS-WSN applications, once deployed, the LS-WSN does not provide a mechanism to evaluate and improve the positions of the initially deployed relay sensors. This paper proposes, based on the growth model of physarum polycephalum and its ability to prune unnecessary links and retain only those deemed useful for food routing, a mechanism for evaluating and optimizing relay sensors in LS-WSNs. Simulation results indicate that the proposed approach significantly improves the network lifetime compared to the initial deployment and that can be a useful approach for LS-WSNs dedicated to Big Data collection. The effectiveness of the proposed technique is demonstrated by experimental results in terms of connectivity and network lifetime.

We study analytically and numerically the impact of the second-neighbor interactions on the propagation of an initial wave-packet through a coupled nonlinear left-handed transmission line. We start with a detailed analysis of the dispersion curve and group velocity. We then use the quasi-discrete approximation with respect to long wavelength transverse perturbations to reduce the nonlinear discrete model of the lattice to a nonlinear Schrödinger equation. A detailed analysis of the product of the group velocity dispersion (P) and the nonlinear term (Q) is presented with respect to the relevant parameters of the system. Intriguingly, we analytically observe that the product can be either simultaneously positive and negative, or positive positive, or even negative negative in different frequency bands. Our numerical results demonstrate that for sufficiently strong second-neighbor coupling, bright backward-wave pulses and dark solitons can coexist at a common frequency, and these travel in opposite directions through the lattice.PACS 05.45,Yv · 04.20.Jb · 42.65.Tg

Nowadays, the cloud computing technology combined with the new generation networks and internet of things facilitate the networking of numerous smart devices. Moreover, the advent of the smart web requires massive data backup from the smart connected devices to the cloud. Unfortunately, the publication of several of these data, such as medical information and financial transactions, could lead to serious privacy breaches, which is becoming the most serious issue in cloud of things. For instance, passive attacks can launched in order to get access to private information. For this reason, several data anonymization techniques have emerged in order to keep data as confidential as possible. However, these different techniques are making the data unusable the most of time. Meanwhile, differential privacy that has been used in a number of cyber physical systems recently emerged as an efficient technique for ensuring the privacy of cloud of things stored data. In this exploratory paper, we study the guarantees of differential privacy of a multi-level anonymization scheme of data graphs. The considered scheme disturbs the structure of the graph by adding false edges, groups the vertices in distinct sets and permutes the vertices in these groups. Particularly, we demonstrated the guarantees that the anonymized data by this algorithm remain exploitable while guaranteeing the anonymity of users.

The fluctuation of ions concentration across the cell membrane of neuron can generate a time varying electromagnetic field. Thus, memristors are used to realize the coupling between the magnetic flux and the membrane potential across the membrane. Such coupling results from the phenomenon of electromagnetic induction in neurons. In this work, we numerically show that the electromagnetic induction phenomenon can firstly suppress chaotic states in a neural setups and secondly enrich neural synchrony in a system of two coupled neurons. By means of the bifurcation diagrams on maximum Lyapunov exponent and interspike interval, we show that increasing in memristor strength delocalizes first, then fully suppresses chaotic states in a single neuron. In a system of two electrically coupled Hindmarsh–Rose-type neurons, we realize that increasing in memristor strength gradually reduces the threshold value of electrical synaptic coupling strength above which a transition to synchronized states is achieved. The transition to synchronized states are determined either by the sign of the maximum transverse Lyapunov exponent or by the magnitude of the synchronization factor. Our results suggest that chaos break in a neurons group by electromagnetic induction phenomenon might automatically release neural synchrony which is involved in information processing and many seizures in the brain.

We consider the nonlinear Schrödinger equation modified by a rational nonlinear term. The model appears in various studies often in the context of the Ginzburg-Landau equation. We investigate modulational instability by means of a linear stability analysis and show how the nonlinear terms affect the growth rate. This analytical result is confirmed by a numerical simulation. The latter analysis shows that breather-like solitons are generated from the instability, and the effects of the nonlinear terms are again clearly seen. Moreover, by employing an auxiliary-equation method we obtain kink and anti-kink soliton as analytical solutions. Our theoretical solution is in good agreement with our numerical investigation.

A theoretical investigation is made to study the properties of dust–acoustic (DA) waves, and corresponding dust–acoustic rogue waves (DARWs) in a magnetized electron-depleted dusty plasma that contains opposite polarity warm dust grains, external magnetic field, and nonextensive distributed ions. By using a multiscale reductive perturbation technique (RPT), the nonlinear Schrodinger (NLS) equation is derived in this model. The effects of magnetic field parameter and nonextensive ions are examined on the profiles of the modulational instability. It is found that with the increase of the magnetic field parameter, the instability growth rate and bandwidth increase (decrease) in the case of positive (negative) nonextensive distributed ions. Moreover, the characteristics (spatial wavelength, width, and amplitude) of nonlinear coherent structures involving the bright and dark solitons, the fundamental and second-order rogue waves are also presented in detail. It is observed that these characteristics are significantly modified by effects of magnetic field parameter and nonextensive ions. The results of the present investigation may be applicable to understanding the characteristics and basic nonlinear structures of magnetized plasma environments both in space and laboratory situations.

The modulational instability of ion-acoustic rogue waves in a three-dimensional magnetized, dissipative plasma consisting of inertial warm positive ions and nonextensive electrons has been theoreti...

This chapter aims to study and solve the perturbed nonlinear Schrödinger (NLS) equation with the power-law nonlinearity in a nano-optical fiber, based upon different methods such as the auxiliary equation method, the Stuart and DiPrima’s stability analysis method, and the bifurcation theory. The existence of the traveling wave solutions is discussed, and their stability properties are investigated through the modulational stability gain spectra. Moreover, the development of the chaotic motions for the systems is pointed out via the bifurcation theory. Taking into account an external periodic perturbation, we have analyzed the chaotic behavior of traveling waves through quasiperiodic route to chaos.

We study the dynamic of modulated solitary wave patterns in alpha-helical proteins going beyond the standard nearest-neighbor interaction by taking into account influence of long-range dipole–dipole interactions of the power low type. In particular, self-trapping of amide-I energy quanta or exciton by the induced phonon deformation of the hydrogen-bonded lattice of peptide groups is capable of generating moving solitary waves following the Davydov soliton model. Under the adiabatic approximation, the discrete biexciton system is reduced to two coupled nonlinear Schrodinger equations where the dispersive coefficient depends on the long-range interactions parameter and the coupled term depends on the nonlinear saturation effects. We perform the linear stability analysis of continuous wave solutions of the coupled system. This analysis reveals that both nonlinear saturation and long-range interaction parameters affect the dispersion area. More new analytical multisoliton, bright and dark soliton solutions-like patterns are obtained using extended auxiliary equation method. We also observe the time behavior of modulating bright soliton-like pattern due to the nonlinear saturation and long-range interactions terms through numerical simulations.

In this paper, a new (2 + 1)-dimensional chemotaxis model is introduced, the focus being the understanding of influences of cooperative mechanisms from traction forces, long-range diffusion to chemotaxis on the dynamical characteristics of waves and their transport. Applying the F-expansion method, three families of new traveling wave solutions of bacterial density and chemoattractant concentration are constructed, including step, dip, and bell-shape wave profiles. The dependence of the conditions of existence of our solutions with respect to the model parameters is fully clarified. We found that traction and long-range diffusion slow down the waves and entail the transport of a small number of particles. Surprisingly, the long-range diffusion increases the thickness of the wave but does not alter its magnitude. Amongst families of solutions constructed, dip waves travel faster may be used to explain fast coordination amongst particles. As they support the transport of large amounts of cells, step waves could explain the transport of particles in high dense media. Intensive numerical simulations corroborate with a pretty much accuracy our theoretical analysis, confirming the robustness of our predictions. Traction, long-range diffusion and chemotaxis deeply affect the wave dynamics, they must be taken into account for a better understanding of chemotaxis systems.

Traveling waves propagating in a two-dimensional advection-chemotactic system in the presence of constant growth and chemoattractant consumption rates are addressed in the present paper. The advection rate is induced by the flow of the fluid within which bacteria are immersed. The traveling wave transformation allows us to reduce the model to a nonlinear ordinary differential equation which is solved by means of an extended F-expansion method. Families of analytical solutions for bacterial density-chemoattractant concentration uncovered exhibit some important characteristics. The critical value of the advection rate in one direction derived shows that it is proportional to the ratio of the wave’s widths in both spatial directions in addition to the advection rate in the orthogonal direction. In this case, solutions are stationary. Above that critical value, the fluid flow rate interacts constructively with the chemotactic motion of bacteria and favors their convergence towards the food source. Below the critical value of the advection rate, the chemotactic motion is counteracted by the flow rate which tends to sweep bacteria and/or swerve the bacterial motion. Further, reducing chemoattractant consumption rate and the medium carrying capacity leads to both wave thickness and amplitude increase, with the consequence of optimizing both spatial repartition and the number of cells carried by the wave. Intensive numerical simulations performed with parameters closed to experimental ones of the original set of model equations corroborate their analytical counterparts with good accuracy, hence paving the way towards the realization of our predictions in current or future experiments

A theoretical model has been developed to study the effects of the gravitational attraction and magnetic field on the waves instabilities as well as the dynamics of dust-acoustic rogue waves (RWs) and the collisions of the envelope dark soliton in self-gravitating non-Maxwellian magnetized electron depleted dusty plasma (EDDP). Using the derivative expansion method, the basic fluid equations of the model are converted to the normal nonlinear Schrodinger equation (NLSE). The modulational instability (MI) analysis is used for determining the regions of (un)stable envelope structures (RWs and envelope solitons). According to gravitational force, a new dispersion relation is obtained and analyzed numerically. It is noted that the presence of gravitational force provides the possibility of a novel purely growing instability mode. Effects of gravitational force and magnetic field on the growth rate of MI and the profile of the RWs in unstable regions and on the phase shifts of the colliding dark solitons in stable regions are discussed in detail. In general, the gravitational force leads to destabilized waves whereas the magnetic field plays the stabilizing role. The present investigation may be of relevance for understanding the mechanism which govern the formation and propagation of modulated DA structures (RWs and envelope solitons) in certain astrophysical objects such as Saturn rings, interstellar medium, dark interstellar clouds, HI and HII regions of galaxies.

We report on the existence and stability of Gaussian-type soliton in the nonlinear Schrodinger (NLS) equation with interplay of cubic–quintic nonlinearity, fourth-order diffraction (FOD) and novel quartic anharmonic parity-time ( $$\mathcal {PT}$$ )-symmetric Gaussian potential. We study numerically the impact of the FOD coefficient on the regions of unbroken/broken linear $$\mathcal {PT}$$ -symmetric phases. In the nonlinear domain, we derive exact soliton solutions of the one-dimensional and two-dimensional cubic–quintic NLS equation with $$\mathcal {PT}$$ -symmetric Gaussian potential and FOD coefficients. Moreover, the stability of the constructed soliton solution is investigated. The results of linear stability analysis are validated by comparison with numerical simulations. Furthermore, we also show that the relative strength of the FOD coefficient influences the direction of the power flow.

Nonlinear excitations of Ca2+ waves are investigated in a two-dimensional cell network with bidirectional paracrine signaling, both in the longitudinal and transversal directions. The semi-discrete...

The competitive effect between electric and magnetic flux couplings is used, in the context of modulational instability, to describe the collective dynamics in a modified Hindmarsh–Rose neural networks. The multiple-scale expansion is utilized to reduce the system to a nonlinear differential-difference equation, whose plane wave solutions are found to be unstable for some values of parameters. Particular interest is given to the influence of changing both the electric and magnetic coupling strengths, and confirmation of analytical results is given via numerical integration of the generic Hindmarsh–Rose model. The model presents a rich variety of spatiotemporal patterns propagating in the network, as the result of the interplay between nonlinear and dispersive effects. The electromagnetic induction appears to be responsible for regular bursting patterns and synchronous states in the network. With increasing the electric coupling, full synchronization is difficult to realize and irregular spatiotemporal patterns of action potentials are predominant.

By introducing the relation between the orders of the two independent small expansion parameters, the amplitude parameter α and the long wavelength parameter β , which are taken in the form β = 0 α n and α = 0 β m , with n and m specified to two particular cases, we derive some new higher order partial differential nonlinear evolution equations for unidirectional shallow water waves under the influence of gravity as well as surface tension. We then show that it is possible to construct one-solution which satisfy these estimates using the Hirota’s bilinear method.

We examine two main behaviors including, firing activity and neuronal synchronization in a Hindmarsh–Rose neural network under magnetic stimulation. We use the theory of bifurcation analysis to seek the control parameters domains in which neuronal activity is set up. The statistical synchronization factor is recorded in order to evaluate the network synchronicity degree. The firing activity is found to be favorable when the values of control parameters, namely the magnetic coupling strength and the external electromagnetic induction, are between low-threshold and upper-threshold and unfavorable elsewhere. Additionally, the increase of electrical coupling strength drastically reduces the domain of control parameters where neuronal synchronization is expected. Our findings suggest that a rich neuronal activity including quiescent state, spiking/bursting regimes, amplitude death phenomenon and global synchronization may be efficiently regulated by electromagnetic induction of intra-and extracellular media of thalamic neurons.

In this paper, we investigate the modulational instability (MI) in an elliptical birefringent positive-negative index coupler. Also, we derive the general expression for instability gain by taking into account cross-phase modulation (XPM), self-steepening (SS) and intrapulse Raman scattering. The instability gain is attained. Using standard linear stability analysis, particularly, we study different configurations of the birefringence with self-steepening or intrapulse Raman scattering effects on MI for both normal and anomalous group velocity dispersion regimes. We observe that the instability gain exhibits significant changes due to the effects of the cross-phase modulation. Finally, we show that efficient control of the MI can be realized by adjusting the self-steepening and intrapulse Raman scattering term, even in the presence of different configurations of the birefringence.

In this paper, we study the dynamics of granular mediums. By comparing each grain of the medium with a ball, and the interaction between various balls like linear elastic interactions, we have established that the dynamics of the system of pendulum used by Fermi–Pasta–Ulam can be governed by a discrete nonlinear Schrodinger equation. Like second advanced, we also established under which conditions this system could be prone to a modulational instability. The long-time dynamics of modulated waves is examined using numerical simulations which are corroborated with analytical results leading to the generation of nonlinear modulated waves which have the shape of a soliton. At the end, we compare our results with what have been carried out previously.

In this paper, the existence and stability of the nonlinear spatial localized modes have been investigated for both self-focusing and self-defocusing in the nonlinear Schrodinger (NLS) equation with interplay of third-order dispersion (TOD), momentum term and complex parity-time $$(\mathcal {PT})$$ -symmetric Scarf potential. The impact of the TOD and momentum coefficients on the regions of unbroken/broken linear $$\mathcal {PT}$$ -symmetric phases has been studied numerically. For the nonlinear case, exact analytical expressions of the localized modes are obtained, respectively, in one- and two-dimensional nonlinear Schrodinger equation with TOD and momentum coefficients. The effects of both TOD and momentum term on the stability/instability structure of these localized modes have also been discussed with the help of linear stability analysis followed by the direct numerical simulation of the governing equation. It was found that the relative strength of the TOD and momentum coefficients can utterly change the direction of the power flow which may be used to control the energy exchange among gain or loss regions.

Dynamics of the positron acoustic waves (PAWs) in magnetoplasmas following Cairns non-thermal distribution is studied on the frameworks of the Korteweg–de Vries (KdV) and modified Korteweg–de Vries (mKdV) equations. The reductive perturbation technique is used to derive the KdV and mKdV equations. Bifurcations of positron acoustic traveling waves of these equations are addressed by employing the bifurcation theory of planar dynamical systems. It is found that the KdV equation supports compressive positron acoustic solitary waves (PASWs), while the mKdV equation supports both compressive and rarefactive PASWs. Using numerical simulations, effect of the nonthermal parameter ( $$\beta $$ ), temperature ratio of hot electron to hot positron ( $$\sigma $$ ), magnetic field ( $$\omega _{c_{1}}$$ ), ratio of hot electron to cold positron concentration ( $$\mu _{e}$$ ), and ratio of hot to cold positron concentration ( $$\mu _{p}$$ ) are discussed on the PASWs solutions of the KdV and mKdV equations. The criterion of chaos for these perturbed equations under the external periodic perturbation are obtained through quasi-periodic route to chaos. It is in fact shown that transition to chaos in our system depends on the frequency $$\omega $$ and the strength of the external periodic perturbation $$f_{0}$$ . These parameters control the dynamic behavior of the PAWs. The relevance of this work may be useful to understand the qualitative changes in the dynamics of perturbed PAWs appearing in auroral acceleration region as well as the astrophysical and laboratory plasma, where static external magnetic field and nonthermal parameter are present.

International audience; In recent years, technological advances and the ever-increasing power of embedded systems have seen the emergence of so-called smart cities. In these cities, application needs are increasingly calling for Large-Scale Wireless Sensor Networks (LS-WSN). However, the design and implementation of such networks pose several important and interesting challenges. These low-cost, low-power devices are characterized by limited computing, memory storage, communication, and battery power capabilities. Moreover, sensors are often required to cooperate in order to route the collected data to a single central node (or sink). The many-to-one communication model that governs dense and widely deployed Wireless Sensor Networks (WSNs) most often leads to problems of network overload and congestion. Indeed, it is easy to show that the closer a node is geographical to the sink, the more data sources it has to relay. This leads to several problems including overloading of nodes close to the sink, high loss rate in the area close to the sink, and poor distribution of power consumption that directly affects the lives of these networks. In this context, we propose a contribution to the problem of LS-WSN energy consumption. We designed a hierarchical 3-tier architecture of LS-WSNs coupled with a modeling of the activities of the different sensors in the network. This architecture that is based on clustering also includes a redeployment function to maintain the topology in case of coverage gaps. The results of the performed simulations show that our architecture maximizes the lifetime than compared solutions.

The Cloud of Things (IoT) that refers to the integration of the Cloud Computing (CC) and the Internet of Things (IoT), has dramatically changed the way treatments are done in the ubiquitous computing world. This integration has become imperative because the important amount of data generated by IoT devices needs the CC as a storage and processing infrastructure. Unfortunately, security issues in CoT remain more critical since users and IoT devices continue to share computing as well as networking resources remotely. Moreover, preserving data privacy in such an environment is also a critical concern. Therefore, the CoT is continuously growing up security and privacy issues. This paper focused on security and privacy considerations by analyzing some potential challenges and risks that need to be resolved. To achieve that, the CoT architecture and existing applications have been investigated. Furthermore, a number of security as well as privacy concerns and issues as well as open challenges, are discussed in this work.

Large Scale Wireless Sensor Networks (LS-WSNs) are Wireless Sensor Networks (WSNs) composed of an impressive number of sensors, with inherent detection and processing capabilities, to be deployed over large areas of interest. The deployment of a very large number of diverse or similar sensors is certainly a common practice that aims to overcome frequent sensor failures and avoid any human intervention to replace them or recharge their batteries, to ensure the reliability of the network. However, in practice, the complexity of LS-WSNs pose significant challenges to ensuring quality communications in terms of symmetry of radio links and maximizing network life. In recent years, most of the proposed LS-WSN deployment techniques aim either to maximize network connectivity, increase coverage of the area of interest or, of course, extend network life. Few studies have considered the choice of a good LS-WSN deployment strategy as a solution for both connectivity and energy consumption efficiency. In this paper, we designed a LS-WSN as a tool for collecting big data generated by smart cities. The intrinsic characteristics of big data require the use of heterogeneous sensors. Furthermore, in order to build a heterogeneous LS-WSN, our scientific contributions include a model of quantifying the kinds of sensors in the network and the multi-level architecture for LS-WSN deployment, which relies on clustering for the big data collection. The results simulations show that our proposed LS-WSN architecture is better than some well known WSN protocols in the literature including Low Energy Adaptive Clustering Hierarchy (LEACH), E-LEACH, SEP, DEEC, EECDA, DSCHE and BEENISH.

A theoretical and numerical study on amplitude modulated positron-acoustic waves (PAWs) in a magnetized four-component space plasma (containing immobile positive ions, inertial cold positrons, and inertia-less hot electrons and positrons following Cairn’s non-thermal distribution function) has been carried out. The reductive perturbation method have been applied to derive the corresponding nonlinear Schrodinger (NLS) equation, whose the nonlinear and dispersion coefficients $Q$ and $P$ are function of the external magnetic field. The criteria for the occurrence of modulational instability (MI) of PAWs is addressed. It is shown that the plasma parameters contribute to enhance substantially the growth rate and the bandwidth of the MI. It is also found from the analysis of the NLS equation that the plasma system under assumption supports the existence of Peregrine solitons and super-rogue waves, whose amplitude are significantly modified by the effects of the external magnetic field, the density ratio of hot positron and cold positron, the density ratio of electron and cold positron, and the non-thermal parameter. Moreover, the various types of localized positron-acoustic excitations exist in the form of bright envelope soliton and dark envelope soliton. It is found that the localized structures’s properties (width and amplitude) are influenced by the presence of magnetic field. The relevance of present study can help researchers to explain the various localized structures and the basic features of PAWs in a magnetized plasmas environments.

Both one-dimensional and two-dimensional localized mode families in parity-time (𝒫𝒯)-symmetric potentials with competing cubic-quintic nonlinearities and higher-order diffraction are reported. In particular, we investigate the role played by the competing nonlinearities and fourth-order diffraction parameter on the beam dynamics in the generalized 𝒫𝒯-symmetric Scarf potentials. The numerical fundamental soliton in competing nonlinearities (1-D double peaked solitons) can be found to be stable around the propagation parameter for exact soliton. A linear stability analysis corroborated by direct numerical simulation reveals that the regions of stability of these solutions can be controlled by tuning the values of the FOD parameters as well as by tuning the sign of the cubic and quintic nonlinearities. In particular, we have shown that the FOD parameter can be used to provide the restoration of the stability of the solitons.

This paper explores the higher-order nonlinear Schrodinger equation with dual power-law nonlinearities and external potential that can depend on space and time coordinates. We constructed exact sol...

A fractional model is proposed to study the effect of heat transfer and magnetic field on the blood flowing inside oscillatory arteries. The flow is due to periodic pressure gradient and the fractional model equations include body acceleration. The proposed velocity and temperature distribution equations are solved using the Laplace and Hankel transforms. The effect of the fluid parameters such as the Reynolds number (Re), the magnetic parameter (M) and the radiation parameter (N) is studied graphically with changing the fractional-order parameter. It is found that the fractional derivative is a valuable tool to control both the temperature and velocity of blood when flow parameters change under treatment, for example. Besides, this work highlights the fact that in the presence of strong magnetic field, blood velocity and temperature reduce. A reversed effect is observed where the applied thermal radiation increase; the velocity and temperature of blood increase. However, the temperature remains high around the artery centerline, which is appropriate during treatment to avoid tissues damage.

Wave propagation in a viscoelastic tube filled with viscous fluid is addressed. We show that the dissipative Navier–Stokes equations can asymptotically be reduced to a pair of nonlinearly coupled complex Ginzburg–Landau equations. Modulational instability is then investigated analytically and numerically. The instability domain, using the growth rate, is shown to be importantly dependent on the vessel relative stiffness and fluid viscosity. A comprehensive analysis is proposed to that effect, which is confirmed by direct numerical simulations. Dissipative trains of impulses are found as the main manifestation of modulational instability and results are recorded for some hemodynamic factors such as the pressure, velocity and vessel cross-section.

In this paper, the existence and stability of solitons in parity-time ( PT )-symmetric optical media characterized by a generic complex hyperbolic refractive index distribution with fourth-order diffraction (FOD) coefficient and higher-order nonlinearities have been investigated. For the linear case, we have demonstrated numerically that, the FOD parameter can alter the PT -breaking points. Exact analytical expressions of the localized modes are obtained respectively, in one and two dimensional nonlinear Schrodinger (NLS) equation with both self-focusing and self-defocusing Kerr, and higher-order nonlinearities for nonlinear case. The effects of both FOD and higher-order nonlinearities on the stability/instability structure of these localized modes have also been discussed with the help of linear stability analysis followed by the direct numerical simulation of the governing equation. Some stable and unstable solutions have been given and, it has been seen how higher-order self-focusing and self-defocusing nonlinearities can influence the stability/instability of the system.